Let f be a differentiable function on R such that f(2)=1, f′(2)=4. Let lim _(x → 0)(f(2+x))^(3 / x)=e^α. Then the number of times the curve y=4 x³-4 x²-4(α-7) x-α meets x-axis is :

Solution

Maths sir · 4 steps

It is a 1^∞ limit, so α = 3f'(2) = 12. Then factor the cubic and count distinct roots.

1. Spot the form. As x → 0, f(2+x) → f(2) = 1 and 3/x → ∞, so the limit has the form 1^∞. For this form, lim gʰ = eˡⁱᵐʰ⁽ᵍ⁻¹⁾.

   α = lim(x → 0) 3(f(2+x) - 1)/x = 3 lim(x → 0) (f(2+x) - f(2))/x = 3f'(2) = 12

2. Build the cubic. With α = 12 we get y = 4x³ - 4x² - 20x - 12 = 4(x³ - x² - 5x - 3). Try x = -1: -1 - 1 + 5 - 3 = 0, so x + 1 is a factor.
3. Factor fully. Dividing gives x³ - x² - 5x - 3 = (x+1)(x² - 2x - 3) = (x+1)²(x-3). The roots are x = -1, a double root where the curve touches the axis, and x = 3.
4. So. The curve meets the x-axis at only two distinct points, (-1, 0) and (3, 0). The answer is 2, option (c).

The idea to keep. For a 1^∞ limit, use eˡⁱᵐʰ⁽ᵍ⁻¹⁾ and look for a derivative hiding inside. When asked how many times a curve meets the axis, count distinct roots.

Answer from the NTA final answer key. Solution by PrepWiser.